The diagram shows part of the curve with equation \(y = x^{\frac{1}{2}} + k^2 x^{-\frac{1}{2}}\), where \(k\) is a positive constant.

(a) Find the coordinates of the minimum point of the curve, giving your answer in terms of \(k\).

The tangent at the point on the curve where \(x = 4k^2\) intersects the y-axis at \(P\).

(b) Find the y-coordinate of \(P\) in terms of \(k\).

The shaded region is bounded by the curve, the x-axis and the lines \(x = \frac{9}{4}k^2\) and \(x = 4k^2\).

(c) Find the area of the shaded region in terms of \(k\).